 where the \(N\times N\)  matrix \(K\)  has scalar entries \(K_{ij}=\Phi ^T(\Omega _2{\bf h}({\bf x}_i))\Phi (\Omega _2{\bf h}({\bf x}_j))\) . The inequality following the convexity of exponential function is important to the heavy-tailed statistics as, by using Theorem 2 in [1]}, we can conclude that
\(\lim _{n_1,n_2\rightarrow \infty }p(f({\bf x}_1)=f({\bf x}_2)=\cdots =f({\bf x}_N)=0)>\mathcal {N}(0_N|0_N,K_{\rm DGP})\:.\) 
